Unavoidable Minors of Large 3-Connected Binary Matroids

Abstract

We show that, for every integer n greater than two, there is a number N such that every 3-connected binary matroid with at least N elements has a minor that is isomorphic to the cycle matroid of K 3, n , its dual, the cycle matroid of the wheel with n spokes, or the vector matroid of the binary matrix (I n | J n &I n ), where J n is the n_n matrix of all ones.

Unavoidable Minors of Large 3-Connected Matroids

“…This paper is a continuation of [1]. In that paper, by building on some new Ramsey-theoretic results for matrices, we distinguished the unavoidable minors in large 3-connected binary matroids.…”

“…Precisely the same matrix results that were applied in [1] to prove Theorem 1.1 will be used here. However, in order to be able to apply these results to arbitrary 3-connected matroids instead of 3-connected binary matroids, we shall introduce the idea of a hamiltonian partial representation.…”

“…This paper is already long, so we shall not repeat the interesting history of results of this type. This history, featuring most notably [3] and [5], may be found in [1]. The bulk of this paper will be concerned with developing the theory of hamiltonian partial representations and manipulating such representations to prove the main theorem.…”

“…This object is a matrix that can be associated with any matroid having a spanning circuit. We will show that this matrix captures enough information about the matroid to enable the main result of [1] to be extended from the relatively well-behaved class of binary matroids to the far-less-tractable class of arbitrary matroids. This paper is already long, so we shall not repeat the interesting history of results of this type.…”

On the interplay between graphs and matroids

“…The first generalization proved was to binary matroids [11]. We denote by J n and 1 the n × n and 1 × n matrices of all ones.…”